The Kinematic Design of Six-bar Linkages Using Polynomial Homotopy Continuation

This dissertation presents the kinematic design of six-bar linkages for function, motion, and path generation by means of polynomial homotopy continuation algorithms. When no link dimensions are specified beforehand, the synthesis formulations for each design objective yield polynomial systems of degrees in the millions and billions, suggesting a large number of solutions. Complete solution sets to these systems have not yet been obtained and is the topic of this dissertation. Function generation for eleven positions is explored in most detail, in particular the Stephenson II and III function generators, for which we calculate multihomogeneous degrees of 264,241,152 and 55,050,240. A numerical reduction using homotopy estimates these systems to have 1,521,037 and 834,441 roots, respectively. For motion generation, the Watt I linkage can be specified for eight positions, producing a system of a multihomogeneous degree over 19 billion. However, for this work we focus on the smaller case of six positions, numerically reducing this system to an estimated 5,735 roots. For path generation we take a different approach. The design of path generators is formulated as RR chains constrained to have a single degree-of-freedom by attaching six-bar function generators to them. This enables us to use our results obtained on Stephenson II and III function generators to create four types of eleven position path generators: the Stephenson I linkage, two types of Stephenson II linkages, and the Stephenson III linkage